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3. Consider the polynomial function h(x) is shown in the graph. Answer the following questions.
a. Create a polynomial function in factored form for
h(x),using the graph and given that h(x) has complex
zeros at x = i and x = - i.
h(x) =

3 Consider the polynomial function hx is shown in the graph Answer the following questions a Create a polynomial function in factored form for hxusing the graph class=

Respuesta :

Answer:

h(x) = (x-i)(x+i)(x+3)

Step-by-step explanation:

There are a few if's here because you should really never trust a graph, but there's also no other way to give a solution to this problem.
Given already in the problem description is two of the roots to h(x), these are the complex roots x=i and x=-i. This immediately tells us that two factors to the polynomial h(x) are (x-i) and (x+i). All we need to do is find the third factor (assuming this is a third degree polynomial). We note that y=0 for x=-3. This implies that (x+3) is a factor to the polynomial h(x) and thereby we have found the factors of h(x).

Combining the factors would render the polynomial [tex]h(x)=x^{3} +3x^{2} +x + 3[/tex].

semsee45

Answer:

[tex]h(x) = (x +3)(x^2+1)[/tex]

Step-by-step explanation:

The shape of the graphed polynomial function is an S-shaped curve, suggesting that it is a cubic function with a degree of 3.

The factored form of a cubic function can be expressed as:

[tex]h(x) = a(x - r_1)(x - r_2)(x - r_3)[/tex]

where:

  • a is the leading coefficient.
  • r₁, r₂ and r₃ are the zeros.

The real zeros of a function are the x-values where the graph of the function intersects or crosses the x-axis. As the graphed polynomial crosses the x-axis at x = -3, then one of the zeros is x = -3.

Given that the other zeros of the function are x = i and x = -i, the factored form of the graphed function h(x) is:

[tex]h(x) = a(x - (-3))(x - i)(x - (-i))[/tex]

[tex]h(x) = a(x +3)(x - i)(x+i)[/tex]

Expand the function:

[tex]h(x) = a(x +3)(x^2+xi-xi-i^2)[/tex]

[tex]h(x) = a(x +3)(x^2-(-1))[/tex]

[tex]h(x) = a(x +3)(x^2+1)[/tex]

To find the value of the leading coefficient, substitute the y-intercept (0, 3) into the function and solve for a:

[tex]\begin{aligned}a(0 +3)(0^2+1)&=3\\a(3)(1)&=3\\3a&=3\\a&=1\end{aligned}[/tex]

Therefore, the equation of the graphed polynomial function is:

[tex]\Large\boxed{\boxed{h(x) = (x +3)(x^2+1)}}[/tex]

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